General Relativity: Metaphysics

Metaphysical Problems of Albert Einstein's General Relativity

Einstein (from Faraday, Maxwell, Lorentz) represented matter as a continuous spherical electromagnetic force field in spacetime. Thus Einstein argues that there is no 'particle' and matter is spherically spatially extended, effectively uniting Space, Time, Matter and Gravity as One thing, spherical fields.

i) Einstein's Relativity is a Theory of a posteriori Effects not a priori Causes, and is founded on Many things (Matter) rather than One thing (Space). However, Metaphysics is founded on the Principle that Reality must be described from One thing that causes and connects the many things (not from our empirical senses of the many things / matter). Unfortunately, Einstein did not know how matter existed in Space and his electromagnetic field theory of matter is Inductive (empirical / a posteriori) and describes effects (of relative motion).

The theory of relativity leads to the same law of motion without requiring any special hypothesis whatsoever as to the structure and behavior of the electron. (Einstein, 1954)

His theory is empirically (a posteriori) founded from observation of how matter 'pushes' other matter around (thus his 'representation' of matter as spherical force fields).

As Ernst Mach insistently pointed out, the Newtonian theory is unsatisfactory in the following respect: if one considers motion from the purely descriptive, not from the causal, point of view, it only exists as relative motion of things with respect to one another.
It compelled Newton to invent a physical space in relation to which acceleration was supposed to exist. This introduction ad hoc of the concept of absolute space, while logically unacceptionable, nevertheless seems unsatisfactory.

Considered logically, concepts are free creations of the human intelligence, tools of thought, which are to serve the purpose of bringing experiences into relation with each other, so that in this way they can be better surveyed. The attempt to become conscious of the empirical sources of these fundamental concepts should show to what extent we are actually bound to these concepts. In this way we become aware of our freedom to create new concepts.

Descartes argued somewhat on these lines: space is identical with extension, but extension is connected with bodies; thus there is no space without bodies and hence no empty space.

It appears to me, therefore, that the formation of the concept of the material object must precede our concepts of time and space. (Albert Einstein, 1954)

Metaphysics, as a true description of Reality, must be based on a priori causes AND these must be united back to one common thing that causes and connects the many things (matter).

ii) Continuous Fields do Not Explain the Discrete Energy Levels of Matter and Light as Determined by Quantum Theory.

The Electric and Magnetic Force Fields were first founded on repeated observations (Induction / a posteriori) of how many trillions of charged 'particles' (electrons and protons) behaved. This explains why the fields were continuous, as many trillions of discrete 'particle' interactions blend together into a continuous force. Thus the continuous field can never describe the real discrete interactions of matter (see Quantum Theory), as Einstein came to realize.

The great stumbling block for the field theory lies in the conception of the atomic structure of matter and energy. For the theory is fundamentally non-atomic in so far as it operates exclusively with continuous functions of space, in contrast to classical mechanics whose most important element, the material point, in itself does justice to the atomic structure of matter. (Einstein, 1954)

iii) Einstein's 'Fields' require 'Particles'.

As Einstein used the empirical/theoretical foundations developed by Faraday, Maxwell and Lorentz he required the existence of a 'Particle' to somehow generate the 'Field' which in turn acted on other 'Particles'.

The special and general theories of relativity, which, though based entirely on ideas connected with the field-theory, have so far been unable to avoid the independent introduction of material points, … the continuous field thus appeared side by side with the material point as the representative of physical reality. This dualism remains even today disturbing as it must be to every orderly mind. (Einstein, 1954)

iv) Einstein's Continuous Field Theory of Matter gives rise to Singularities and Infinite Fields.

The Maxwell equations in their original form do not, however, allow such a description of particles, because their corresponding solutions contain a singularity. Theoretical physicists have tried for a long time (1936), therefore, to reach the goal by a modification of Maxwell's equations. These attempts have, however, not been crowned with success. What appears certain to me, however, is that, in the foundations of any consistent field theory the particle concept must not appear in addition to the field concept. The whole theory must by based solely on partial differential equations and their singularity-free solutions. (Einstein, 1954)

As Wolff explains (see the Wave Structure of Matter), the equation for a scalar spherical wave give rise to a finite wave-amplitude at the wave-center (consistent with observation) whereas spherical vector electromagnetic fields tend to infinity as the radius tends to zero.

v) Einstein Rejects both 'Particles' and Motion.

While Einstein correctly rejected the point 'particle' concept of matter, he assumed that Motion only applied to 'particles' (a common error!) thus he also rejected the concept of Motion, and represented matter as spherical force fields. The error is twofold; firstly, he did not consider the (wave) Motion of Space itself, and secondly, he should have realized that to measure forces we must first measure the change in Motion of a 'particle', thus Motion is a priori to forces (i.e. Force = dE/dx).

Since the theory of general relativity implies the representation of physical reality by a continuous field, the concept of particles or material points cannot play a fundamental part, nor can the concept of motion. (Einstein, 1954)

The past 80 years of Modern Physics now strongly suggest that neither the 'Particle' nor the continuous electromagnetic force 'Field' is a complete description of Reality. If we reject both the 'Particle' and the 'Field', then what remains is Motion. Hence we open another path to explore - the study of Space as a wave medium for wave Motion - and that the Spherical Wave Motion of Space explains both the 'particle' (wave-center) and 'forces' (interaction of spherical In and out Waves) properties of matter.

vi) Einstein Assumed Matter Caused Space Rather than the Wave-Motion of Space Causing Matter.

Einstein was profoundly influenced by Mach;

Mach, in the nineteenth century, was the only one who thought seriously of the elimination of the concept of space, in that he sought to replace it by the notion of the totality of the instantaneous distances between all material points. (He made this attempt in order to arrive at a satisfactory understanding of inertia.) (Einstein, 1954)

Because we only observe the motion of matter relative to all the other matter in the universe, thus Einstein thought that matter, rather than Space, must be the central perspective for representing Reality. Thus Einstein's Relativity is empirically (a posteriori) founded from observing the motion of matter relative to other matter.
Einstein is empirically correct, and at the same time this was his error because Metaphysics (and thus Reality) is not founded on empirical observations. As Newton wrote;

And so instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in Philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. (Newton, 1687)

Further, Lorentz's assumption of an Absolute Space is the foundation for the Lorentz transformations and thus for Einstein's Relativity.

I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. (Lorentz, The Theory of the Electron, 1906)

Einstein choose to ignore Space / Aether and work with relative motions of matter to other matter, with matter being represented by spherical fields.

The electromagnetic fields are not states of a medium, and are not bound down to any bearer, but they are independent realities which are not reducible to anything else. (Albert Einstein, Leiden Lecture, 1920)
In other words, is there an ether which carries the field; the ether being considered in the undulatory state, for example, when it carries light waves? The question has a natural answer: Because one cannot dispense with the field concept, it is preferable not to introduce in addition a carrier with hypothetical properties. (Albert Einstein, 1950)

However, if the particle and the continuous electromagnetic field it generates are both merely ideas, human approximations to reality, then we must consider Space and its Properties. We return to Lorentz's foundation of One thing Space, and its properties as a wave medium (vibrations) and replace the spherical particle & field with the spherical wave Motion of Space.
Thus it now seems likely that the conception of the field theory of matter misled Einstein, and yet Einstein also realised that there must somehow be a Space that interconnects matter.

Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it. (Albert Einstein, Leiden Lecture, 1920)

vii) Einstein Never United the Electromagnetic & Gravitational Fields into a Unified Field Theory for Matter.

Einstein's Relativity requires both an Electromagnetic Force Field to explain Charge, and a Gravitational Field (matter curves space-time) to explain Mass. He tried and failed throughout his life to unite these two fields into one (and to remove the 'particle' concept from them).

But the idea that there exist two structures of space independent of each other, the metric-gravitational and the electromagnetic, was intolerable to the theoretical spirit. We are prompted to the belief that both sorts of field must correspond to a unified structure of space. (Einstein, 1954)

viii) Einstein's 'Curvature of the Four Dimensional Space-Time Continuum'

The concept of the 'curvature of space' is a mathematical construction of Einstein's general relativity. The Wave Structure of Matter suggests that Space is not 'curved', instead (for gravitational forces) the wave-density of Space varies dependent upon the nearby proximity of matter, and this causes a variation in the velocity of waves/light which changes the ellipsoidal shape of matter and causes the curved path of matter and light in Space. And this caused Einstein considerable problems (it took him ten years to work out the ellipsoidal geometry for gravity/general relativity!)

But the path (of general relativity) was thornier than one might suppose, because it demanded the abandonment of Euclidean geometry. This is what we mean when we talk of the 'curvature of space'. The fundamental concepts of the 'straight line', the 'plane', etc., thereby lose their precise significance in physics. (Albert Einstein, 1954)

Further, the four dimensional space-time continuum simply means that three spatial dimensions and a time dimension are required to define the motion of bodies and the path of light in three dimensional Space.

The non-mathematician is seized by a mysterious shuddering when he hears of 'four-dimensional' things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum. Space is a three-dimensional continuum. ... Similarly, the world of physical phenomena is naturally four dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x, y, z, and the time co-ordinate t. (Albert Einstein, 1954)

The inseparability of time and space emerged in connection with electrodynamics, or the law of propagation of light.
With the discovery of the relativity of simultaneity, space and time were merged in a single continuum in a way similar to that in which the three dimensions of space had previously merged into a single continuum. Physical space was thus extended to a four dimensional space which also included the dimension of time. The four dimensional space of the special theory of relativity is just as rigid and absolute as Newton's space. (Albert Einstein, 1954)

Now this is very important, for it is this 'curvature' that largely led to Einstein's early fame. It was the prediction by Einstein that light curved as it grazed the sun (subsequently confirmed by observation during a solar eclipse on the 29th May 1919) that resulted in his General Theory of Relativity becoming widely accepted and very famous. His general principle seems to be correct though, matter does determine the geometric properties of Space;

According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. (Einstein, 1954)

Editor: Haselhurst

General Relativity: Theoretical Foundations

Theoretical Foundations of Albert Einstein's General Relativity

(On Accelerated Motion and Gravitation. 1915)

General Relativity extends Special Relativity to include accelerated Motion, and the relationship between Force, Mass, Acceleration, Time and Space / Dimension.

Prior to Albert Einstein's Theory of Relativity, gravity had been described by Newton as a Force which acts instantly between particles (action-at-a-distance). This was very unsatisfactory as no explanation was given to explain this (apparently magical!) action-at-distance.
Einstein very cleverly used Lorentz's Theory of the Electron (1900) to propose a completely new way of looking at Gravity. Lorentz had discovered that for accelerated electrons the experimental results only made sense if it was assumed that the electron (and its electromagentic forces) changed shape with acceleration (from a sphere to a squashed ellipsoid). Einstein thus realised that acceleration was related to the changing ellipsoidal geometry of a sphere. And so by replacing the motion of particles in Space and Time (Newton) with spherical (ellipsoidal) fields in Space-Time, Einstein was able to relate the geometry of Space-Time with the gravitational acceleration of matter. Thus Einstein's Theory of Gravitation is not a theory of forces, but rather a theory of geometry, and that matter 'curves' the 'four dimensional Space-Time Continuum'. i.e. The earth has a curved path around the sun not because of any forces, but because the sun curves the geomerty of space-time which the earth is moving through.

Now for a little more detail. If the above ideas of Einstein's are to work it requires that Inertial Mass (it takes force to accelerate a body) and Gravitational Mass (a body accelerates in a gravitational field) must be equivalent (as Einstein relates these two types of mass as explained above).

Inertial Mass and Gravitational Mass

Now for the principle of the conservation of mass. Mass is defined by the resistance that a body opposes to its acceleration (inert mass). It is also measured by the weight of the body (gravity mass). That these two radically different definitions lead to the same value for the mass of a body is in itself an astonishing fact. ... According to the principle - namely, that masses remain unchanged under any physical or chemical changes - the mass appeared to be the essential (because unvarying) quality of matter.
Physicists accepted this principle up to a few decades ago. But it proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy principle. ... We might say that the principle of the conservation of energy, having preciously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass - and holds the field alone. (Albert Einstein, 1946)

It is an unsatisfactory feature of classical mechanics that in its fundamental laws the same mass constant appears in two different roles, namely as 'inertial mass' in the law of motion, and as 'gravitational mass' in the law of gravitation. (Albert Einstein, 1936)

The Equivalence of Inertial Mass and Gravitational Mass

Let us now consider a simple example of this equivalence. Imagine standing in a room, the room existing in Space away from any stars or other massive bodies. We would be weightless in the Space as there would be no gravitational effect.
Now if we imagine the room being accelerated upwards, (relative to the floor), at 9.8m/s, as the occupant of the room, we would not be able to tell if we are being accelerated or if we are in the Earth's gravitational field.
Further, if there is a rope attached to an object hanging from the ceiling of the room, the tension in the rope could be due either to the inertia caused by accelerating the room, or to the object's weight due to its mass in a gravitational field. This is the empirical equivalence of gravitational and inertial mass.

The establishment of this general principle of relativity is made easier by a fact of experience that has long been known, namely, that the weight and the inertia of a body are controlled by the same constant (equality of inertial and gravitational mass). This hasty consideration suggests that a general theory of relativity must supply the laws of gravitation, and the consistent following up of the idea has justified our hopes. But the path was thornier than one might suppose, because it demanded the abandonment of Euclidean geometry. This is what we mean when we talk of the 'curvature of space'. The fundamental concepts of the 'straight line', the 'plane', etc., thereby lose their precise significance in physics.
In the general theory of relativity the doctrine of space and time, or kinematics, no longer figures as a fundamental independent of the rest of physics. The geometrical behaviour of bodies and the motion of clocks rather depend on gravitational fields which in their turn are produced by matter. (Albert Einstein, 1919)

The principle of the equivalence of inertial and gravitational mass could now be formulated quite clearly as follows: in a homogenous gravitation field all motions take place in the same way as in the absence of a gravitational field in relation to uniformly accelerated co-ordinate system. (Albert Einstein, 1934)

There is no reason to exclude the possibility of interpreting this behaviour as the effect of a 'true' gravitational field (principle of equivalence of inertial/gravitational mass). This interpretation implies that A is an 'inertial system', even though it is accelerated with respect to another inertial system. (Albert Einstein, 1950)

And so we see that Albert Einstein based his mathematics for gravitation, on the fact that Matter in an accelerated reference frame (Inertial Mass) behaved the same as Matter in a gravitational field (Gravitational Mass). (Principle of Equivalence.)
Thus if we know the Lorentz transformation for moving with a constant velocity, (which require linear transformations of the co-ordinate system) then we can calculate how the Lorentz transformation would change if the reference frame is now accelerated.

.. the theory of gravitation is based on the principle of equivalence discussed above and rests on the following consideration: according to the theory of special relativity, light has a constant velocity of propagation. If a light ray in a vacuum starts from a point, designated by the co-ordinates X1, X2, and X3 in a three dimensional co-ordinate system, at the time X4; it spreads as a spherical wave and reaches a neighbouring point (X1+dX1, X2+dX2, X3+dX3) at the time X4+dX4. Introducing the velocity of light, c, we write the expression:
dX1^2 + dX2^2 + dX3^2 = (c.dX4)^2
This expression represents an objective relation between neighbouring space time points in four dimensions, and it holds for all inertial systems, provided the co-ordinate transformations are restricted to those of special relativity. The relation loses this form, however, if arbitrary continuous transformations of the co-ordinates are admitted in accordance with the principle of general relativity. (The equations expressing the laws of nature must be covariant with respect to all continuous transformations of the co-ordinates. This is the principle of general relativity.) (Albert Einstein, 1954)

Albert Einstein is thus forced to use a curved (non-linear) co-ordinate system (rather than linear as per Special Relativity and the Lorentz Transformations), which he found from the work of Gauss and Riemann (called symmetrical tensors.)

In order to account for the equality of inert and gravitational mass within the theory it necessary to admit non-linear transformations of the four co-ordinates. Mathematics suggests an answer which is based of the fundamental investigations of Gauss and Riemann. (Albert Einstein, 1954)

To introduce this non-linear transformation, it was necessary for Albert Einstein to adjust the velocity of light dependent upon the energy density (gravitational field) of Space.

(Special relativity is founded) on the basis of the law of the constancy of the velocity of light. But the general theory of relativity cannot retain this law. On the contrary, we arrived at the result that according to this latter theory the velocity of light must always depend on the co-ordinates when a gravitational field is present. (Albert Einstein, 1954)

Over the past 80 years it has become clear that Einstein's General Relativity does accurately describe the motion of matter due to the influence of Gravity. However, Einstein failed to develop a unified field theory of matter, and it is now considered likely that a continuous field theory of matter does not correctly represent reality. More recent theories (String Theory, Wave Structure of Matter) show some promise of solving these problems.


1. - Philosophy and Metaphysics of Einstein's Special and General Relativity